Entire Exam 2 notes

Chapter Four: Introductory Probability

Introduction to Probability

• Probability pertains to the likelihood of an event occurring based on repeated experiments under identical conditions.

• The speaker expresses a passion for probability, highlighting the contrast with students' general discomfort with the topic.

The Basics of Probability

• Probability Notation:

  • Notated as P(E), where "E" is any event (e.g., P(friends getting married), P(dying on the way home)).

• Probability Range:

  • Probability values must fall between 0 and 1:

    • 0 = impossible event

    • 1 = certain event

• Example of improper probability: No negative probabilities (e.g., -15% is invalid).

Mutually Exclusive Events

• Definition: Events that cannot happen simultaneously.

  • Example: Flipping a coin results in either heads or tails - not both.

• In contexts like sports, if one team wins, the other team can't win simultaneously (e.g., Eagles vs Chiefs in the Super Bowl).

• Probability Calculation:

  • For mutually exclusive events, the sum of probabilities equals 1.

  • Example: Coin flip: P(heads) + P(tails) = 0.5 + 0.5 = 1.0.

  • Other example: If P(Chiefs winning) = 60%, then P(Eagles winning) = 40%.

Independent Events

• Definition: The occurrence of one event does not affect the occurrence of another event.

  • Example: Someone eating at Chick-fil-A doesn't affect someone else eating at McDonald's.

• Probability Calculation:

  • For independent events, P(E1 and E2) = P(E1) * P(E2).

  • Example: If P(E1) = 0.65 and P(E2) = 0.45, then P(E1 and E2) = 0.65 * 0.45 = 0.2925.

Calculating Probability for Events

• Probability of Either Event:

  • Calculation formula: P(E1 or E2) = P(E1) + P(E2) - P(E1 and E2).

  • Problem-solving involves defining probabilities and accounting for overlaps.

• Example: For the probability of getting a heads or tails in two coin flips, consider possibilities to avoid double counting.

Sample Space

• Definition: The set of all possible outcomes of an experiment.

  • Example: Flipping a coin twice results in these outcomes: HH, HT, TH, TT.

• To calculate possibilities, multiply outcomes by the number of flips (e.g., 2 outcomes for the first flip x 2 for the second = 4 total outcomes).

Relative Frequency Assessment

• Definition: An empirical way to estimate probabilities based on past occurrences.

  • Example: A Starbucks manager might say the probability of a customer ordering a caffeinated drink is 1570 out of 2250 total sales.

• Umbrella issue: Insufficient models may miss key data affecting probabilities, such as time of day for drink sales.

Subjective Probability Assessment

• Definition: A personal estimation of an event's chance occurring, often without statistical backing, like predicting weather or game outcomes based on gut feeling.

Practical Example of Drawing Cards

• Problem statement: What is the probability of drawing a red card or an ace from a standard deck of 52 cards?

  • Count of red cards = 26 (13 hearts, 13 diamonds).

  • Count of aces = 4 (one from each suit).

  • Adjust for double counting the 2 red aces: Use P(Red) + P(Ace) - P(Red Ace).

  • Example calculations for clarity on adjustments in shared probabilities.

Conclusion

• The chapter covers foundational aspects of probability essential for further statistical understanding.

• The speaker encourages relatable examples to reinforce concepts while preparing for the exam.

Introduction

• Class discussion focuses on probability and relevant formulas for upcoming test.

Independent Events

• Probability of two independent events:

  • Given probabilities: P(E1) = 0.6, P(E2) = 0.45.

  • Formula for both events occurring: P(E1 and E2) = P(E1) * P(E2).

  • Example calculation: 0.6 * 0.45 = 0.27.

• Next question involves finding the probability of either event occurring:

  • Formula: P(E1 or E2) = P(E1) + P(E2) - P(E1 and E2).

  • Calculation example:

    • P(E1 or E2) = 0.6 + 0.45 - 0.27 = 0.78.

Mutually Exclusive vs Independent Events

• For mutually exclusive events, P(E1 and E2) = 0 because they cannot occur together.

• Venn diagram comparison:

  • Mutually exclusive: no overlap between events.

  • Independent: overlap is possible; hence overlap must be subtracted to avoid double counting when calculating probabilities.

Example: Drawing Cards

• Probability focused on drawing cards:

  • P(Ace) = 4/52 = 7.7%.

  • P(Red card) = 26/52 = 50%.

  • P(Red card and Ace): Only two possible cards (Ace of Diamonds and Ace of Hearts) = 2/52 = 3.8%.

  • Calculation of P(Red card or Ace): 26 + 4 - 2 = 28/52 = 53.8%.

Conditional Probability

• Definition: Probability of an event given another event has occurred (dependent events).

• Relevant for situations like sports outcomes where one event affects the probability of another:

  • Example: Calculating probability that Chiefs win Super Bowl given they win AFC Championship.

  • Formula: P(E1|E2) = P(E1 and E2) / P(E2).

Example Calculation

• Example: Given Aces and Red Cards from a deck:

  • Event E1: Drawing Ace of Diamonds.

  • Event E2: Drawing a Red Card.

  • Conditional probability calculation where outcomes are dependent based on previous draws.

Test Preparation

• Important elements to focus on for the test:

  • Probability questions involving independent and mutually exclusive events.

  • Conditional probability and practical card drawing scenarios.

Sampling with/without Replacement

• Highlight concept differences:

  • Sampling with replacement: probabilities remain unchanged from draw to draw.

  • Sampling without replacement: Probability changes depending on previous outcomes.

Conclusion

• Key concepts of probability relevant for the test include:

  • Independent vs mutually exclusive events, conditional probability, and understanding sample space.

  • Practice calculations and probabilities with provided examples.

Final Comments

• Reminder about the significance of understanding when sampling with or without replacement impacts probability calculations.

Introduction to Probability Distributions

• Types of Probability Distributions:

  • Binomial Distribution

  • Poisson Distribution

  • Hypergeometric Distribution

• Focus Today: Binomial Probability Distribution

Discrete Variables

• Definition of Discrete Variables:

  • Countable number of unique values (e.g., 0, 1, 2, 3, etc.)

  • Cannot take on fractional values (e.g., 4.75 is not valid)

• Continuous Variables:

  • Can assume uncountable values (discussed in Chapter 6)

Binomial Probability Distribution

Example: Coin Flipping Experiment

• Experiment: Flip a coin 3 times

• Sample Space:

  • 3 heads, 2 heads, 1 head, 0 heads

• Probability Distribution for Heads on 3 Flips:

  • Exactly 0 heads: 1 way (TTT) = 1/8 = 0.125 (12.5%)

  • Exactly 1 head: 3 ways (HTT, THT, TTH) = 3/8 = 0.375 (37.5%)

  • Exactly 2 heads: 3 ways (HHT, HTH, THH) = 3/8 = 0.375 (37.5%)

  • Exactly 3 heads: 1 way (HHH) = 1/8 = 0.125 (12.5%)

Importance of Probability Calculations

• Highlighted Material: Important for the in-class portion of the exam

Graphing Probability Distribution

• Creating Graphs in Excel:

  • Insert recommended charts, choose vertical bar chart

  • Adjust axes labels to match probabilities of 0, 1, 2, 3 heads

  • Set y-axis to show probabilities from 0 to 1 for clarity

Expected Value of a Discrete Variable

• Expected Value (Mean/Average):

  • Formula: E(X) = Σ (x_i * P(x_i))

• Calculation Example for Coin Flips:

  • Expected values calculated from probabilities for 0, 1, 2, 3 heads

  • Result: Expected Value = 1.5

Variance and Standard Deviation

Variance Calculation

• Formula: Variance = Σ [(x_i - E(X))^2 * P(x_i)]

• Calculation steps for 0, 1, 2, 3 heads using probabilities from earlier

Standard Deviation

• Determination: Standard Deviation = √(Variance)

• Example Result: √0.75 = 0.866

Characteristics of the Binomial Distribution

• Properties Required for Binomial Distribution:

  1. n identical trials

  2. Two mutually exclusive outcomes (success or failure)

  3. Trials are independent

  4. Probability of success (p) is constant

• Probability of Failure (q): q = 1 - p

Binomial Probability Formula

• Formula for Probability:

  • P(X = x) = (n! / (x! (n - x)!)) * p^x * q^(n-x)

  • Example: Calculate probabilities for set n and x values, using factorials

Combinations Formula

• Combination Formula: C(n, x) = n! / (x!(n - x)!)

• Example: Determine combinations for choosing successes in trials (e.g., choosing 2 successes from 3 trials)

Practical Application Using Excel

• Utilizing Excel Functions:

  • Using BINOM.DIST for exact probabilities

  • Entering the number of successes, trials, and probabilities for calculations

Cumulative Probabilities

• Definition: Probability that X is less than or equal to a certain value

• Calculation in Excel: BINOM.DIST function with cumulative set to TRUE

Probability Greater Than or Less Than

• Greater Than: P(X > x) = 1 - P(X ≤ x)

• Less Than: P(X < x) = P(X ≤ x-1)

Binomial Distribution Graphs

• Effect of p (Probability of Success):

  • As p approaches 0.5, distribution becomes more symmetric regardless of n (number of trials)

• Visual Representation: Graphs illustrating probability distributions based on varying p values and n

Summary and Next Steps

• Preparation for Next Class: Introduction to Poisson Distribution

• Note for Test Preparation: Formulas and values to be provided

• Reminder: Watch videos for further assistance with upcoming homework assignments.

Discrete Probability Distributions

Overview

• Focus on Poisson and Hypergeometric distributions.

• Assignments and solutions available on Canvas; classwork assistance provided.

Poisson Distribution

• Definition: Used for modeling count data (e.g., number of successes in a time segment).

• Examples:

  • Scholarship offers for high school athletes from universities.

  • Number of emergency calls in an hour at a hospital.

  • Potholes in a specified highway segment.

Key Characteristics

• Count data consists of whole numbers and cannot be fractional (e.g., 1 or 2 scholarships).

• Assumption: Average number of successes in one segment (denoted by lambda, λ).

• Trials are independent, with consistent probability of success across segments.

Application Example

• Scenario: Average bank customers arriving is 16 per hour (λ = 16).

• Calculating Probabilities:

  • Use formula: P(x) = (λ * t)^x * e^(-λ * t) / x!

  • Where:

    • λ = average number of successes

    • t = time segment

    • x = number of successes

Practical Calculation Using Excel

• Setup: Input average, time, and number of successes in designated cells.

• Formula for Exact Probability: =POISSON.DIST(x, λ*t, FALSE) for exact occurrences; =POISSON.DIST(x, λ*t, TRUE) for cumulative probability.

Assumptions and Limitations

• Both expected value and variance in Poisson distribution are equal to λ * t.

• Assumes mean and variance equality, which often doesn’t hold in real scenarios.

• Poisson is less practical; consider using the negative binomial distribution instead for systems where they differ.

Hypergeometric Distribution

• Usage: Best for dependent trials where the probability of success changes from trial to trial.

• Overview:

  • Used for scenarios where items are chosen without replacement.

  • Important in binary outcomes but with dependent trials.

Hypergeometric Formula

• Formula: P(X = x) = [C(N - K, n - k) * C(K, k)] / C(N, n)

  • Where:

    • N = population size

    • K = total number of successes in the population

    • n = sample size

    • k = number of observed successes in the sample

Example Scenario

• Context: Firm downsizing with a set population of employees (30 total).

• Layoffs randomly select 10 employees; success defined as female staff members.

• Calculate probabilities for different outcomes (e.g., eight women laid off).

Excel Calculations for Hypergeometric Distribution

• Set required population values and sample sizes in Excel for easy probability calculations.

• Use formula in Excel to streamline calculations without needing to compute manually.

Comparison of Distributions

• Binomial Distribution: Applicable for independent trials with binary data.

• Poisson Distribution: For independent trials with count data.

• Hypergeometric Distribution: For dependent trials with either count or binary data.

• Distinguish use cases based on the independence/dependency of trials and nature of the data.

Introduction

• Objective: To cover Concepts of Continuous Probability Distributions

• Schedule: Finish Chapter 6 and go through exam review this week.

Overview of Probability Distributions

• Discrete Probability Distributions reviewed in Chapter 5: Whole number variables (e.g., binomial, Poisson)

• Continuous Probability Distributions (Chapter 6): Three key types to discuss:

  • Normal Distribution

  • Uniform Distribution

  • Exponential Distribution

Importance of Normal Distribution

• The normal distribution is the most commonly used distribution in statistics

  • Variables that are normally distributed allow for various statistical techniques

• Examples of continuous variables:

  • Age can be measured in decimal, e.g., 42.331507 years.

  • Income can also take decimal values.

Understanding Continuous Variables

• Continuous variables can take any real number within a range.

• Probability of observing an exact value (e.g., age) is approximately zero.

• Typically, we look for probabilities within a range (e.g., the probability of being between 42 and 43 years old).

Normal Distribution Properties

• Use of the normal distribution formula:

[ f(x) = \frac{1}{\sigma \sqrt{2 \pi}} e^{-\frac{(x-\mu)^2}{2\sigma^2}}
] (Not required for calculations, better for graphing)

• Standard properties:

  • Mean = Median = Mode at the peak,

  • The curve is symmetric about the mean.

  • 50% of data points fall below the mean.

Standard Deviation and Variance

• Examples of how the standard deviation affects the normal curve:

  • Smaller standard deviations result in a steeper curve.

  • Larger standard deviations lead to a flatter curve.

• Sample mean is an unbiased estimate of the population mean.

• Variance of the sample mean: (\sigma^2/n), where (n) = sample size.

• Larger sample sizes yield less variable sampling distributions.

Z-Scores

• Calculation of Z-scores to standardize any normal distribution variable:

  • [ Z = \frac{X - \mu}{\sigma}
    ]

  • Example given with response times in a medical emergency is shown.

• Z-score interpretation in context of response times:

  • Z-score helps determine how many standard deviations away from the mean a particular response time is.

Normal Distribution and Excel

• Instructions for using Excel to calculate probabilities:

  • Use NORM.DIST() function for cumulative probabilities.

• Example showed calculating the areas under the curve using Excel.

Empirical Rule Recap

• 68% of data falls within one standard deviation,

• 95% within two, and

• 99.7% within three standard deviations around the mean.

Uniform Distribution

• Definition: Continuous probability where all outcomes are equally likely within a specified range.

• Formula: ( f(x) = \frac{1}{b-a} ) if ( x ) is between ( a ) and ( b ) (inclusive)

• Example applied to pine tree diameter growth between 1 and 4 inches.

Cumulative Probability for Uniform Distribution

• To find a probability that ( x ) is greater than a certain value, use cumulative probabilities. This will yield:

  • ( P(X > x) = 1 - P(X \leq x) )

• Example with pine tree growth assessments explained.

Exponential Distribution

• Used to measure time between events (e.g., time until the next customer arrives).

• Defining Characteristics:

  • Mean ( \mu = \frac{1}{\lambda} )

  • Standard Deviation equals Mean: also ( \sigma = \frac{1}{\lambda} )

• Visual representation noted for exponential decay.

Summary and Review

• Emphasized the importance of understanding different probability distributions before diving into applying them, including how they relate to real-world situations.

Overview of Exam Instructions

• The exam includes a take-home portion that consists of multiple-choice and short-answer questions.

• The in-class portion must be completed on Tuesday.

• Students are required to submit the take-home exam during the in-class portion.

Exam Format

Take-Home Portion

• Starts with question 20 and includes multiple-choice questions

• Ends with short-answer questions

• Utilizes chapters 5 and 6 materials primarily

In-Class Portion

• Consists mainly of multiple-choice questions

• Related to chapters 4, 5, and 6 assignments

• Students should bring calculators

Exam Study Material

Excel Worksheets

• Essential for both sections of the exam, especially chapters 5 and 6

• Utilize provided worksheets on Canvas for calculations

• Worksheets contain information necessary for problems related to binomial, Poisson, and normal distributions

Key Chapters and Content

Chapter 5

• Reference the binomial and hypergeometric distributions

• Important to understand how to manipulate given Excel sheets for problem-solving

Chapter 6

• Focus on the normal distribution and how to calculate probabilities

• Key to understand cumulative probability formulas for exponential distribution

Key Formulas and Concepts

Binomial Distribution

• Formula for Expected Value: E(X) = n * p

• Variance Formula: Var(X) = n * p * (1 - p)

Poisson and Exponential Distributions

• Key Characteristics: Mean and variance are equal for Poisson; for exponential, the mean and standard deviation are equal.

• Mean (Expected Value) of Exponential Distribution: 1/lambda

Probability Problems

Typical Question Formats

• Students will be expected to calculate probabilities based on given statistics using the appropriate distribution formulas

• Be able to differentiate between mutually exclusive and independent events.

Example Question

• Given n = 8 and p = 0.37 in a binomial distribution scenario:

  • Calculate expected value and variance

Usage of Technology

• Students can use Excel or Google Sheets for calculations

• Caution against using external human resources for answers

Preparing for the Exam

• Complete highlighted problems from assignments on Canvas

• Review textbook examples pertaining to the take-home exam

Additional Exam Tips

• Bring necessary materials including calculators and answer sheets

• Stay organized with answer sheets and questions

• Revisit class notes and Excel worksheets before the exam

Questions and Clarifications

• Open for Q&A at the end of the review session

• Ensure copy of the take-home exam is obtained either in class or via email if watching online.

• Prepare any questions in advance to maximize the effectiveness of the Q&A session.